9–61. Evaluate and simplify y'.

y = e^6x...

Question

9–61. Evaluate and simplify y'.

y = e^6x sin x

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says to find the derivative of the function Y is equal to E rates to the quantity of 3 X in quantity multiplied by cosine X. We're given 4 possible choices as our answers. For choice A, we have E rated to the quantity of 3 X in 20 multiplied by the quantity of 3 cosine x plus sine x. For choice B, we have E rated to the quantity of 3 X in quantity, multiplied by the quantity of 3 cosine x minus sine X. For choice C, we have E rates to the quantity of 3 X in quantity, multiplied by the quantity of 3 sin x minus cosine X, and for choice D, we have Erates to the quantity of 3 X in quantity, multiplied by the quantity of 3 sin x plus cosine X. Now we're asked to find a derivative of our function Y. And in order to take a derivative of our function Y, we're going to need to apply our product rule. So we call for the product rule that we have Y going to be equal to U multiplied by V. Plus, you multiplied by V, where we have rewritten Y as being equal to U multiplied by V. So here we need to identify what U is and what V is in our expression. So, if we look at our equation, we can identify you as being equal to E raised to the quantity of 3 X. And V as being equal to cosine X. So we need to take a derivative of each of these quantities. So taking a derivative of U with respect to X will have U. And this is going to be equal to 3, multiplied by E raised to the quantity of 3 X. And then we'll have V is equal to minus sin x. So now we need to just substitute all these expressions into our equation for Y. We will have Y. is equal to U, which is the quantity of 3, multiplied by E rates to the quantity of 3 X. In quantity. And that gets multiplied by V, which is cosine X. Then we'll have plus U, which is E raises to the quantity of 3 X. In quantity, multiplied by the, which is the quantity of minus X. In quantity So just simplifying this expression, we can factor out an E raised to the quantity of 3 X out of both terms. So we'll have E raised to the quantity of 3 X in quantity. That's going to be multiplying the quantity of what's left, which is going to be 3, cosine X. Minus 9 X. And so this is the answer that we are looking for. And if we compare our calculated answer to the answer choices, we see that the correct choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

9–61. Evaluate and simplify y'.

y = e^6x sin x

Key Concepts

Differentiation

Product Rule

Chain Rule

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